



Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
The spring 2003 exam 2 for the ece 434: random processes course at the university of illinois at urbana-champaign. The exam covers various topics related to markov processes, brownian motion, poisson processes, and gaussian processes. Students are required to solve problems on finding q matrices, first-order probability distributions, markov properties, martingales, expected values, and probabilities.
Typology: Exams
1 / 6
This page cannot be seen from the preview
Don't miss anything!




Spring 2003 Exam 2
Monday, April 14, 2003
Name:
Score:
Total: (40 pts.)
Problem 1 (9 points) Let X be a stationary continuous-time Markov process with the transition rate diagram shown.
(a) Write down the Q matrix and find the first-order probability distribution π. (So π is a probability vector, representing the distribution of Xt for each t.)
(b) Let X^2 denote the process X^2 = (X t^2 : t ∈ IR). Is X^2 a Markov process? Justify your answer.
(c) Is (Xt : t ≥ 0) a martingale? Justify your answer.
Problem 3 (6 points) Let (Nt : t ≥ 0) be a Poisson process with rate λ > 0. Find P [N 2 ≥ 3 |N 4 = 4].
Problem 4 (8 points) Let Z = (Zt : t ∈ IR) be a mean zero, stationary Gaussian process with RZ (τ ) = e−|τ^ |^ and let V =
∫ (^) ∞ 0 e
−tZtdt.
(a) Find E[V 2 ].
(b) Find P [V ≥ 3].